| L(s) = 1 | − 2·2-s − 3·3-s − 5·5-s + 6·6-s + 35·7-s + 8·8-s + 10·10-s + 4·11-s − 154·13-s − 70·14-s + 15·15-s − 16·16-s + 26·17-s + 121·19-s − 105·21-s − 8·22-s + 166·23-s − 24·24-s + 308·26-s + 27·27-s + 12·29-s − 30·30-s − 235·31-s − 12·33-s − 52·34-s − 175·35-s + 419·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 0.316·10-s + 0.109·11-s − 3.28·13-s − 1.33·14-s + 0.258·15-s − 1/4·16-s + 0.370·17-s + 1.46·19-s − 1.09·21-s − 0.0775·22-s + 1.50·23-s − 0.204·24-s + 2.32·26-s + 0.192·27-s + 0.0768·29-s − 0.182·30-s − 1.36·31-s − 0.0633·33-s − 0.262·34-s − 0.845·35-s + 1.86·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9162927496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9162927496\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T - 1315 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 77 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 26 T - 4237 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 121 T + 7782 T^{2} - 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 166 T + 15389 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 235 T + 25434 T^{2} + 235 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 419 T + 124908 T^{2} - 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 128 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 291 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 442 T + 91541 T^{2} - 442 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 276 T - 72701 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 706 T + 293057 T^{2} - 706 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 442 T - 31617 T^{2} + 442 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 531 T - 18802 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1036 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 427 T - 206688 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1317 T + 1241450 T^{2} + 1317 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1188 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 38 T - 703525 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 866 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87713952329416067207207360120, −11.61950477293276484062328678542, −11.29388806835248978747075763921, −10.82010833471902580530082153800, −10.06218919034266967451534970691, −9.734022344399903819236098406845, −9.393044688553666764458669810538, −8.514965950920390535166107807346, −8.230545394014920469010600323499, −7.39565069092781994958228640997, −7.33928122255142596372191694885, −7.02265654612836698085555850481, −5.47888869017974315913259207689, −5.40029395788707811072567008745, −4.67804807902066959973565144784, −4.50175231443416026445264715724, −3.16308830038606081532489946337, −2.34502760183776069194409509629, −1.44063598419253818038512407389, −0.50357107832474907511694803993,
0.50357107832474907511694803993, 1.44063598419253818038512407389, 2.34502760183776069194409509629, 3.16308830038606081532489946337, 4.50175231443416026445264715724, 4.67804807902066959973565144784, 5.40029395788707811072567008745, 5.47888869017974315913259207689, 7.02265654612836698085555850481, 7.33928122255142596372191694885, 7.39565069092781994958228640997, 8.230545394014920469010600323499, 8.514965950920390535166107807346, 9.393044688553666764458669810538, 9.734022344399903819236098406845, 10.06218919034266967451534970691, 10.82010833471902580530082153800, 11.29388806835248978747075763921, 11.61950477293276484062328678542, 11.87713952329416067207207360120
