L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 7-s + 3·8-s − 2·9-s + 10-s − 4·11-s − 12-s − 14-s − 15-s − 16-s − 7·17-s + 2·18-s − 6·19-s + 20-s + 21-s + 4·22-s + 3·24-s + 25-s − 5·27-s − 28-s − 29-s + 30-s + 2·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.69·17-s + 0.471·18-s − 1.37·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.962·27-s − 0.188·28-s − 0.185·29-s + 0.182·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4527477103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4527477103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.228420525937747407839093809818, −7.76744383019049868255394843704, −6.92272368448513913881745203543, −6.01476220591566084542852976669, −4.98887384857355165993248986219, −4.50519087712402184526752918094, −3.66953672996679460039064911820, −2.57636942202727521077428791229, −1.93965576995848206884547556730, −0.36546946424659758161868586229,
0.36546946424659758161868586229, 1.93965576995848206884547556730, 2.57636942202727521077428791229, 3.66953672996679460039064911820, 4.50519087712402184526752918094, 4.98887384857355165993248986219, 6.01476220591566084542852976669, 6.92272368448513913881745203543, 7.76744383019049868255394843704, 8.228420525937747407839093809818