L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s + 4·13-s + 16-s + 3·17-s − 18-s + 6·19-s + 4·22-s + 7·23-s + 24-s − 4·26-s − 27-s + 4·29-s − 5·31-s − 32-s + 4·33-s − 3·34-s + 36-s + 2·37-s − 6·38-s − 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.37·19-s + 0.852·22-s + 1.45·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.898·31-s − 0.176·32-s + 0.696·33-s − 0.514·34-s + 1/6·36-s + 0.328·37-s − 0.973·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255351163\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255351163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85430054217221537190759160969, −7.31585753955669542453351207857, −6.64991795936521023773199737401, −5.61046196584100776406721198578, −5.47636927080261135543181837631, −4.42262663475980796888713975185, −3.32045365251011806484838908984, −2.73454218038125663847549354289, −1.42850528946506500775439331102, −0.71157567193332827471999343542,
0.71157567193332827471999343542, 1.42850528946506500775439331102, 2.73454218038125663847549354289, 3.32045365251011806484838908984, 4.42262663475980796888713975185, 5.47636927080261135543181837631, 5.61046196584100776406721198578, 6.64991795936521023773199737401, 7.31585753955669542453351207857, 7.85430054217221537190759160969