| L(s) = 1 | + 3-s + 5-s − 4·9-s − 6·11-s + 2·13-s + 15-s − 2·17-s + 8·19-s − 2·23-s − 8·25-s − 6·27-s + 2·29-s + 3·31-s − 6·33-s − 8·37-s + 2·39-s − 11·41-s − 13·43-s − 4·45-s + 8·47-s − 2·51-s − 3·53-s − 6·55-s + 8·57-s − 61-s + 2·65-s − 15·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 4/3·9-s − 1.80·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 1.83·19-s − 0.417·23-s − 8/5·25-s − 1.15·27-s + 0.371·29-s + 0.538·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s − 1.71·41-s − 1.98·43-s − 0.596·45-s + 1.16·47-s − 0.280·51-s − 0.412·53-s − 0.809·55-s + 1.05·57-s − 0.128·61-s + 0.248·65-s − 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 159 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 253 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−8.533732283680664436577677949803, −8.200815921439230341000192803341, −7.60489439089374269628216770320, −7.54781745966044508964304292064, −7.10258384005820657997189200229, −6.49150780981797374005224583921, −6.06228527587588333025655071844, −5.77802355164132056466793269052, −5.36861685929996625705144193005, −5.16606883853046922716908066146, −4.73395582670959533051629313518, −4.05786882343667755727723394743, −3.52129479636843063116126499966, −3.20069528201449284083376971016, −2.69119096772723637478398714994, −2.59311966921292282545756144899, −1.71948920196424384451510979032, −1.45218580027003231569425818843, 0, 0,
1.45218580027003231569425818843, 1.71948920196424384451510979032, 2.59311966921292282545756144899, 2.69119096772723637478398714994, 3.20069528201449284083376971016, 3.52129479636843063116126499966, 4.05786882343667755727723394743, 4.73395582670959533051629313518, 5.16606883853046922716908066146, 5.36861685929996625705144193005, 5.77802355164132056466793269052, 6.06228527587588333025655071844, 6.49150780981797374005224583921, 7.10258384005820657997189200229, 7.54781745966044508964304292064, 7.60489439089374269628216770320, 8.200815921439230341000192803341, 8.533732283680664436577677949803
