| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s − 4·10-s + 2·11-s + 6·12-s + 4·14-s + 4·15-s + 5·16-s − 6·18-s + 6·20-s − 4·21-s − 4·22-s + 4·23-s − 8·24-s + 3·25-s + 4·27-s − 6·28-s + 4·29-s − 8·30-s + 8·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 1.41·18-s + 1.34·20-s − 0.872·21-s − 0.852·22-s + 0.834·23-s − 1.63·24-s + 3/5·25-s + 0.769·27-s − 1.13·28-s + 0.742·29-s − 1.46·30-s + 1.43·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996511690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996511690\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.991187076116031992991246591110, −8.921301098169327697114145379832, −8.494540218643585055453234845819, −8.380694225009657755698764156911, −7.58668995396088903980732021247, −7.49591110959358096011606772824, −6.87334003114123372684617601851, −6.70960634929088182198961383864, −6.23634273714901971750450507631, −6.01650133882870636178002912544, −5.20817516547221745938831936882, −4.98104918234492355914653584014, −4.10544537411848441830162153941, −3.81596213784528510356757406468, −3.15706569440454412149446973725, −2.79793511663247812338419173423, −2.29468685460817520592942856162, −2.02261168009596405856499840659, −1.00749381042922416750058467572, −0.888788568762048489544398883449,
0.888788568762048489544398883449, 1.00749381042922416750058467572, 2.02261168009596405856499840659, 2.29468685460817520592942856162, 2.79793511663247812338419173423, 3.15706569440454412149446973725, 3.81596213784528510356757406468, 4.10544537411848441830162153941, 4.98104918234492355914653584014, 5.20817516547221745938831936882, 6.01650133882870636178002912544, 6.23634273714901971750450507631, 6.70960634929088182198961383864, 6.87334003114123372684617601851, 7.49591110959358096011606772824, 7.58668995396088903980732021247, 8.380694225009657755698764156911, 8.494540218643585055453234845819, 8.921301098169327697114145379832, 8.991187076116031992991246591110
